Relating properties of a ring and its ring of row and column finite matrices

Mackey and Ornstein proved that if R is a semi-simple ring then the ring of row and column finite matrices over R (RCFMGamma(R)) is a Baer ring for an y infinite set Gamma. A ring with identity is a Baer ring if every left (eq uivalent every right) annihilator is generated by an idempotent. This resul t is discussed in Kaplansky's book, "Rings of Operators." This result is of course decades old. Here we prove that the converse is true. The proof is long and we develop techniques which allow us to obtain results of a more m odern flavor about RCFMGamma(R) where R is a perfect or semi-primary ring. Finally, we obtain good enough results on annihilators in RCFM(Z) to show t hat this ring is coherent. (C) 2001 Academic Press.